Stretched exponentials from superstatistics

Distributions exhibiting fat tails occur frequently in many different areas of science. A dynamical reason for fat tails can be a so-called superstatistics, where one has a superposition of local Gaussians whose variance fluctuates on a rather large spatio-temporal scale. After briefly reviewing this concept, we explore in more detail a class of superstatistics that hasn’t been subject of many investigations so far, namely superstatistics for which a suitable power βη of the local inverse temperature β is χ2-distributed. We show that η>0 leads to power-law distributions, while η<0 leads to stretched exponentials. The special case η=1 corresponds to Tsallis statistics and the special case η=-1 to exponential statistics of the square root of energy. Possible applications for granular media and hydrodynamic turbulence are discussed.